An Episodic History of Mathematics potx

An Episodic History of
Mathematics
Mathematical Culture through Problem Solving
by Steven G. Krantz
September 23, 2006
To Marvin J. Greenberg, an inspiring teacher.
iii
Preface
Together with philosophy, mathematics is the oldest academic dis-
cipline known to mankind. Today mathematics is a huge and complex
enterprise, far b eyond the ken of any one individual. Those of us who
choose to study the subject can only choose a piece of it, and in the end
must specialize rather drastically in order to make any contribution to
the evolution of ideas.
An important development of twenty-first century life is that mathe-
matical and analytical thinking have permeated all aspects of our world.
We all need to understand the spread of diseases, the likelihood that we
will contract SARS or hepatitis. We all must deal with financial matters.
Finally, we all must deal with computers and databases and the Internet.
Mathematics is an integral part of the theory and the operating systems
that make all these computer systems work. Theoretical mathematics is
used to design automobile bodies, to plan reconstructive surgery proce-
dures, and to analyze prison riots. The modern citizen who is unaware
of mathematical thought is lacking a large part of the equipment of life.
Thus it is worthwhile to have a book that will introduce the student
to some of the genesis of mathematical ideas. While we cannot get into
the nuts and bolts of Andrew Wiles’s solution of Fermat’s Last Theorem,
we can instead describe some of the stream of thought that created the
problem and led to its solution. While we cannot describe all the sophis-
ticated mathematics that go es into the theory b ehind black holes and
modern cosmology, we can instead indicate some of Bernhard Riemann’s

ideas about the geometry of space. While we cannot describe in spe-
cific detail the mathematical research that professors at the University
of Paris are performing to day, we can instead indicate the development
of ideas that has led to that work.
Certainly the modern school teacher, who above all else serves as a
role model for his/her students, must b e conversant with mathematical
thought. As a matter of course, the teacher will use mathematical ex-
amples and make mathematical allusions just as examples of reasoning.
Certainly the grade school teacher will seek a book that is broadly ac-
cessible, and that sp eaks to the level and interests of K-6 students. A
book with this audience in mind should serve a good purpose.
iv
Mathematical history is exciting and rewarding, and it is a signifi-
cant slice of the intellectual pie. A good education consists of learning
different metho ds of discourse, and certainly mathematics is one of the
most well-developed and important mo des of discourse that we have.
The purpose of this book, then, is to acquaint the student with
mathematical language and mathematical life by means of a number of
historically important mathematical vignettes. And, as has already been
noted, the book will also serve to help the prospective school teacher to
become inured in some of the important ideas of mathematics—both
classical and modern.
The focus in this text is on doing—getting involved with the math-
ematics and solving problems. This book is unabashedly mathematical:
The history is primarily a device for feeding the reader some doses of
mathematical meat. In the course of reading this book, the neophyte
will become involved with mathematics by working on the same prob-
lems that Zeno and Pythagoras and Descartes and Fermat and Riemann
worked on. This is a book to be read with pencil and paper in hand, and
a calculator or computer close by. The student will want to experiment,

to try things, to b ecome a part of the mathematical pro cess.
This history is also an opportunity to have some fun. Most of the
mathematicians treated here were complex individuals who led colorful
lives. They are interesting to us as people as well as scientists. There are
wonderful stories and anecdotes to relate about Pythagoras and Galois
and Cantor and Poincar´e, and we do not hesitate to indulge ourselves in
a little whimsy and gossip. This device helps to bring the subject to life,
and will retain reader interest.
It should be clearly understood that this is in no sense a thorough-
going history of mathematics, in the sense of the wonderful treatises of
Boyer/Merzbach [BOM] or Katz [KAT] or Smith [SMI]. It is instead a col-
lection of snapshots of aspects of the world of mathematics, together with
some cultural information to put the mathematics into perspective. The
reader will pick up history on the fly, while actually doing mathematics—
developing mathematical ideas, working out problems, formulating ques-
tions.
And we are not shy about the things we ask the reader to do. This
book will be accessible to students with a wide variety of backgrounds
v
and interests. But it will give the student some exposure to calculus, to
number theory, to mathematical induction, cardinal numbers, cartesian
geometry, transcendental numbers, complex numbers, Riemannian ge-
ometry, and several other exciting parts of the mathematical enterprise.
Because it is our intention to introduce the student to what mathemati-
cians think and what mathematicians value, we actually prove a number
of important facts: (i) the existence of irrational numbers, (ii) the exis-
tence of transcendental numbers, (iii) Fermat’s little theorem, (iv) the
completeness of the real number system, (v) the fundamental theorem of
algebra, and (vi) Dirichlet’s theorem. The reader of this text will come
away with a hands-on feeling for what mathematics is about and what

mathematicians do.
This book is intended to be pithy and brisk. Chapters are short, and
it will be easy for the student to browse around the book and select topics
of interest to dip into. Each chapter will have an exercise set, and the
text itself will be peppered with items labeled “For You to Try”. This
device gives the student the opportunity to test his/her understanding
of a new idea at the moment of impact. It will be both rewarding and
reassuring. And it should keep interest piqued.
In fact the problems in the exercise sets are of two kinds. Many of
them are for the individual student to work out on his/her own. But
many are labeled for class discussion. They will make excellent group
projects or, as appropriate, term papers.
It is a pleasure to thank my editor, Richard Bonacci, for enlisting me
to write this book and for providing decisive advice and encouragement
along the way. Certainly the reviewers that he engaged in the writing
process provided copious and detailed advice that have turned this into
a more accurate and useful teaching tool. I am grateful to all.
The instructor teaching from this book will find grist for a num-
ber of interesting mathematical projects. Term papers, and even honors
projects, will be a natural outgrowth of this text. The book can be used
for a course in mathematical culture (for non-majors), for a course in the
history of mathematics, for a course of mathematics for teacher prepa-
ration, or for a course in problem-solving. We hope that it will help to
bridge the huge and demoralizing gap between the technical world and
the humanistic world. For certainly the most important thing that we
do in our society is to communicate. My wish is to communicate math-
ematics.
SGK
St. Louis, MO
Table of Contents

Preface
1 The Ancient Greeks 1
1.1 Pythagoras . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Introduction to Pythagorean Ideas . . . . . . . . 1
1.1.2 Pythagorean Triples . . . . . . . . . . . . . . . . 7
1.2 Euclid 10
1.2.1 Introduction to Euclid . . . . . . . . . . . . . . . 10
1.2.2 The Ideas of Euclid . . . . . . . . . . . . . . . . . 14
1.3 Archimedes . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.3.1 The Genius of Archimedes . . . . . . . . . . . . . 21
1.3.2 Archimedes’s Calculation of the Area of a Circle . 24
2 Zeno’s Paradox and the Concept of Limit 43
2.1 The Context of the Paradox? . . . . . . . . . . . . . . . 43
2.2 The Life of Zeno of Elea . . . . . . . . . . . . . . . . . . 44
2.3 Consideration of the Paradoxes . . . . . . . . . . . . . . 51
2.4 Decimal Notation and Limits . . . . . . . . . . . . . . . 56
2.5 Infinite Sums and Limits . . . . . . . . . . . . . . . . . . 57
2.6 Finite Geometric Series . . . . . . . . . . . . . . . . . . . 59
2.7 Some Useful Notation . . . . . . . . . . . . . . . . . . . . 63
2.8 Concluding Remarks . . . . . . . . . . . . . . . . . . . . 64
3 The Mystical Mathematics of Hypatia 69
3.1 Introduction to Hypatia . . . . . . . . . . . . . . . . . . 69
3.2 What is a Conic Section? . . . . . . . . . . . . . . . . . . 78
vii
viii
4 The Arabs and the Development of Algebra 93
4.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . 93
4.2 The Development of Algebra . . . . . . . . . . . . . . . . 94
4.2.1 Al-Khowˆarizmˆı and the Basics of Algebra . . . . . 94
4.2.2 The Life of Al-Khwarizmi . . . . . . . . . . . . . 95

4.2.3 The Ideas of Al-Khwarizmi . . . . . . . . . . . . . 100
4.2.4 Omar Khayyam and the Resolution of the Cubic . 105
4.3 The Geometry of the Arabs . . . . . . . . . . . . . . . . 108
4.3.1 The Generalized Pythagorean Theorem . . . . . . 108
4.3.2 Inscribing a Square in an Isosceles Triangle . . . . 112
4.4 A Little Arab Number Theory . . . . . . . . . . . . . . . 114
5 Cardano, Ab el, Galois, and the Solving of Equations 123
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.2 The Story of Cardano . . . . . . . . . . . . . . . . . . . 124
5.3 First-Order Equations . . . . . . . . . . . . . . . . . . . 129
5.4 Rudiments of Second-Order Equations . . . . . . . . . . 130
5.5 Completing the Square . . . . . . . . . . . . . . . . . . . 131
5.6 The Solution of a Quadratic Equation . . . . . . . . . . . 133
5.7 The Cubic Equation . . . . . . . . . . . . . . . . . . . . 136
5.7.1 A Particular Equation . . . . . . . . . . . . . . . 137
5.7.2 The General Case . . . . . . . . . . . . . . . . . . 139
5.8 Fourth Degree Equations and Beyond . . . . . . . . . . . 140
5.8.1 The Brief and Tragic Lives of Abel and Galois . . 141
5.9 The Work of Ab el and Galois in Context . . . . . . . . . 148
6 Ren´e Descartes and the Idea of Coordinates 151
6.0 Introductory Remarks . . . . . . . . . . . . . . . . . . . 151
6.1 The Life of Ren´e Descartes . . . . . . . . . . . . . . . . . 152
6.2 The Real Number Line . . . . . . . . . . . . . . . . . . . 156
6.3 The Cartesian Plane . . . . . . . . . . . . . . . . . . . . 158
6.4 Cartesian Coordinates and Euclidean Geometry . . . . . 165
6.5 Coordinates in Three-Dimensional Space . . . . . . . . . 169
7 The Invention of Differential Calculus 177
7.1 The Life of Fermat . . . . . . . . . . . . . . . . . . . . . 177
7.2 Fermat’s Method . . . . . . . . . . . . . . . . . . . . . . 180
ix

7.3 More Advanced Ideas of Calculus: The Derivative and the
TangentLine 183
7.4 Fermat’s Lemma and Maximum/Minimum Problems . . 191
8 Complex Numb ers and Polynomials 205
8.1 A New Number System . . . . . . . . . . . . . . . . . . . 205
8.2 Progenitors of the Complex Number System . . . . . . . 205
8.2.1 Cardano . . . . . . . . . . . . . . . . . . . . . . . 206
8.2.2 Euler 206
8.2.3 Argand . . . . . . . . . . . . . . . . . . . . . . . 210
8.2.4 Cauchy 212
8.2.5 Riemann . . . . . . . . . . . . . . . . . . . . . . . 212
8.3 Complex Number Basics . . . . . . . . . . . . . . . . . . 213
8.4 The Fundamental Theorem of Algebra . . . . . . . . . . 219
8.5 Finding the Roots of a Polynomial . . . . . . . . . . . . 226
9 Sophie Germain and Fermat’s Last Problem 231
9.1 Birth of an Inspired and Unlikely Child . . . . . . . . . . 231
9.2 Sophie Germain’s Work on Fermat’s Problem . . . . . . 239
10 Cauchy and the Foundations of Analysis 249
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 249
10.2 Why Do We Need the Real Numbers? . . . . . . . . . . . 254
10.3 How to Construct the Real Numbers . . . . . . . . . . . 255
10.4 Properties of the Real Number System . . . . . . . . . . 260
10.4.1 Bounded Sequences . . . . . . . . . . . . . . . . . 261
10.4.2 Maxima and Minima . . . . . . . . . . . . . . . . 262
10.4.3 The Intermediate Value Property . . . . . . . . . 267
11 The Prime Numbers 275
11.1 The Sieve of Eratosthenes . . . . . . . . . . . . . . . . . 275
11.2 The Infinitude of the Primes . . . . . . . . . . . . . . . . 278
11.3 More Prime Thoughts . . . . . . . . . . . . . . . . . . . 279
12 Dirichlet and How to Count 289

12.1 The Life of Dirichlet . . . . . . . . . . . . . . . . . . . . 289
12.2 The Pigeonhole Principle . . . . . . . . . . . . . . . . . . 292
x
12.3 Other Types of Counting . . . . . . . . . . . . . . . . . . 296
13 Riemann and the Geometry of Surfaces 305
13.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 305
13.1 How to Measure the Length of a Curve . . . . . . . . . . 309
13.2 Riemann’s Method for Measuring Arc Length . . . . . . 312
13.3 The Hyperbolic Disc . . . . . . . . . . . . . . . . . . . . 316
14 Georg Cantor and the Orders of Infinity 323
14.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . 323
14.2 What is a Number? . . . . . . . . . . . . . . . . . . . . . 327
14.2.1 An Uncountable Set . . . . . . . . . . . . . . . . 332
14.2.2 Countable and Uncountable . . . . . . . . . . . . 334
14.3 The Existence of Transcendental Numbers . . . . . . . . 337
15 The Number Systems 343
15.1 The Natural Numbers . . . . . . . . . . . . . . . . . . . 345
15.1.1 Introductory Remarks . . . . . . . . . . . . . . . 345
15.1.2 Construction of the Natural Numbers . . . . . . . 345
15.1.3 Axiomatic Treatment of the Natural Numbers . . 346
15.2 The Integers . . . . . . . . . . . . . . . . . . . . . . . . . 347
15.2.1 Lack of Closure in the Natural Numbers . . . . . 347
15.2.2 The Integers as a Set of Equivalence Classes . . . 348
15.2.3 Examples of Integer Arithmetic . . . . . . . . . . 348
15.2.4 Arithmetic Properties of the Integers . . . . . . . 349
15.3 The Rational Numbers . . . . . . . . . . . . . . . . . . . 349
15.3.1 Lack of Closure in the Integers . . . . . . . . . . . 349
15.3.2 The Rational Numbers as a Set of Equivalence
Classes 350
15.3.3 Examples of Rational Arithmetic . . . . . . . . . 350

15.3.4 Subtraction and Division of Rational Numbers . . 351
15.4 The Real Numbers . . . . . . . . . . . . . . . . . . . . . 351
15.4.1 Lack of Closure in the Rational Numbers . . . . . 351
15.4.2 Axiomatic Treatment of the Real Numb ers . . . . 352
15.5 The Complex Numbers . . . . . . . . . . . . . . . . . . . 354
15.5.1 Intuitive View of the Complex Numbers . . . . . 354
15.5.2 Definition of the Complex Numbers . . . . . . . . 354
xi
15.5.3 The Distinguished Complex Numbers 1 and i . . 355
15.5.4 Algebraic Closure of the Complex Numbers . . . 355
16 Henri Poincar´e, Child Prodigy 359
16.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . 359
16.2 Rubber Sheet Geometry . . . . . . . . . . . . . . . . . . 364
16.3 The Idea of Homotopy . . . . . . . . . . . . . . . . . . . 365
16.4 The Brouwer Fixed Point Theorem . . . . . . . . . . . . 367
16.5 The Generalized Ham Sandwich Theorem . . . . . . . . 376
16.5.1 Classical Ham Sandwiches . . . . . . . . . . . . . 376
16.5.2 Generalized Ham Sandwiches . . . . . . . . . . . 378
17 Sonya Kovalevskaya and Mechanics 387
17.1 The Life of Sonya Kovalevskaya . . . . . . . . . . . . . . 387
17.2 The Scientific Work of Sonya Kovalevskaya . . . . . . . . 393
17.2.1 Partial Differential Equations . . . . . . . . . . . 393
17.2.2 A Few Words About Power Series . . . . . . . . . 394
17.2.3 The Mechanics of a Spinning Gyroscope and the
Influence of Gravity . . . . . . . . . . . . . . . . . 397
17.2.4 The Rings of Saturn . . . . . . . . . . . . . . . . 398
17.2.5 The Lam´e Equations . . . . . . . . . . . . . . . . 399
17.2.6 Bruns’s Theorem . . . . . . . . . . . . . . . . . . 400
17.3 Afterward on Sonya Kovalevskaya . . . . . . . . . . . . . 400
18 Emmy Noether and Algebra 409

18.1 The Life of Emmy Noether . . . . . . . . . . . . . . . . . 409
18.2 Emmy Noether and Abstract Algebra: Groups . . . . . . 413
18.3 Emmy Noether and Abstract Algebra: Rings . . . . . . . 418
18.3.1 The Idea of an Ideal . . . . . . . . . . . . . . . . 419
19 Methods of Proof 423
19.1 Axiomatics . . . . . . . . . . . . . . . . . . . . . . . . . 426
19.1.1 Undefinables . . . . . . . . . . . . . . . . . . . . . 426
19.1.2 Definitions . . . . . . . . . . . . . . . . . . . . . . 426
19.1.3 Axioms . . . . . . . . . . . . . . . . . . . . . . . 426
19.1.4 Theorems, ModusPonendoPonens, and ModusTol
lens 427
xii
19.2 Proof by Induction . . . . . . . . . . . . . . . . . . . . . 428
19.2.1 Mathematical Induction . . . . . . . . . . . . . . 428
19.2.2 Examples of Inductive Proof . . . . . . . . . . . . 428
19.3 Proof by Contradiction . . . . . . . . . . . . . . . . . . . 432
19.3.1 Examples of Pro of by Contradiction . . . . . . . . 432
19.4 Direct Proof . . . . . . . . . . . . . . . . . . . . . . . . . 434
19.4.1 Examples of Direct Proof . . . . . . . . . . . . . . 435
19.5 Other Methods of Proof . . . . . . . . . . . . . . . . . . 437
19.5.1 Examples of Counting Arguments . . . . . . . . . 437
20 Alan Turing and Cryptography 443
20.0 Background on Alan Turing . . . . . . . . . . . . . . . . 443
20.1 The Turing Machine . . . . . . . . . . . . . . . . . . . . 445
20.1.1 An Example of a Turing Machine . . . . . . . . . 445
20.2 More on the Life of Alan Turing . . . . . . . . . . . . . . 446
20.3 What is Cryptography? . . . . . . . . . . . . . . . . . . . 448
20.4 Encryption by Way of Affine Transformations . . . . . . 454
20.4.1 Division in Modular Arithmetic . . . . . . . . . . 455
20.4.2 Instances of the Affine Transformation Encryption 457

20.5 Digraph Transformations . . . . . . . . . . . . . . . . . . 461
References 437
Chapter 1
The Ancient Greeks and the
Foundations of Mathematics
1.1 Pythagoras
1.1.1 Introduction to Pythagorean Ideas
Pythagoras (569–500 B.C.E.) was both a person and a society (i.e., the
Pythagoreans). He was also a p olitical figure and a mystic. He was
sp ecial in his time because, among other reasons, he involved women as
equals in his activities. One critic characterized the man as “one tenth
of him genius, nine-tenths sheer fudge.” Pythagoras died, according to
legend, in the flames of his own school fired by political and religious
bigots who stirred up the masses to protest against the enlightenment
which Pythagoras sought to bring them.
As with many figures from ancient times, there is little specific that
we know about Pythagoras’s life. We know a little about his ideas and
his school, and we sketch some of these here.
The Pythagorean society was intensely mathematical in nature, but
it was also quasi-religious. Among its tenets (according to [RUS]) were:
• To abstain from beans.
• Not to pick up what has fallen.
• Not to touch a white cock.
• Not to break bread.
• Not to step over a crossbar.
1
2 Chapter 1: The Ancient Greeks
• Not to stir the fire with iron.
• Not to eat from a whole loaf.
• Not to pluck a garland.

• Not to sit on a quart measure.
• Not to eat the heart.
• Not to walk on highways.
• Not to let swallows share one’s roof.
• When the pot is taken off the fire, not to leave the mark
of it in the ashes, but to stir them together.
• Not to look in a mirror beside a light.
• When you rise from the bedclothes, roll them together
and smooth out the impress of the body.
The Pythagoreans embodied a passionate spirit that is remarkable
to our eyes:
Bless us, divine Number, thou who generatest gods
and men.
and
Number rules the universe.
The Pythagoreans are remembered for two monumental contribu-
tions to mathematics. The first of these was to establish the impor-
tance of, and the necessity for, proofs in mathematics: that mathemati-
cal statements, especially geometric statements, must b e established by
way of rigorous proof. Prior to Pythagoras, the ideas of geometry were
generally rules of thumb that were derived empirically, merely from ob-
servation and (occasionally) measurement. Pythagoras also introduced
the idea that a great body of mathematics (such as geometry) could be
1.1 Pythagoras 3
a
b
Figure 1.1. The fraction
b
a
.

derived from a small number of postulates. The second great contribu-
tion was the discovery of, and proof of, the fact that not all numbers are
commensurate. More precisely, the Greeks prior to Pythagoras believed
with a profound and deeply held passion that everything was built on
the whole numbers. Fractions arise in a concrete manner: as ratios of
the sides of triangles (and are thus commensurable—this antiquated ter-
minology has today been replaced by the word “rational”)—see Figure
1.1.
Pythagoras proved the result that we now call the Pythagorean theo-
rem. It says that the legs a, b and hypotenuse c of a right triangle (Figure
1.2) are related by the formula
a
2
+ b
2
= c
2
. ()
This theorem has perhaps more proofs than any other result in
mathematics—over fifty altogether. And in fact it is one of the most
ancient mathematical results. There is evidence that the Babylonians
and the Chinese knew this theorem nearly 1000 years before Pythago-
ras.
In fact one proof of the Pythagorean theorem was devised by Pres-
ident James Garfield. We now provide one of the simplest and most
4 Chapter 1: The Ancient Greeks
a
b
c
Figure 1.2. The Pythagorean theorem.

classical arguments. Refer to Figure 1.3.
Proof of the Pythagorean Theorem:
Observe that we have four right triangles and a square packed into a
larger square. Each triangle has legs a and b, and we take it that b>a.
Of course, on the one hand, the area of the larger square is c
2
. On the
other hand, the area of the larger square is the sum of the areas of its
component pieces.
Thus we calculate that
c
2
= (area of large square)
= (area of triangle) + (area of triangle) +
(area of triangle) + (area of triangle) +
(area of small square)
=
1
2
· ab +
1
2
· ab +
1
2
· ab +
1
2
· ab +(b −a)
2

=2ab +[a
2
− 2ab + b
2
]
1.1 Pythagoras 5
a
b
c
c
c
c
a
a
a
b
b
b
Figure 1.3
6 Chapter 1: The Ancient Greeks
= a
2
+ b
2
.
That proves the Pythagorean theorem.
For You to Try: If c = 10 and a = 6 then can you determine what b
must be in the Pythagorean theorem?
Other proofs of the Pythagorean theorem will be explored in the exer-
cises, as well as later on in the text.

Now Pythagoras noticed that, if a = 1 and b = 1, then c
2
= 2. He
wondered whether there was a rational number c that satisfied this last
identity. His stunning conclusion was this:
Theorem: There is no rational number c such
that c
2
=2.
Proof: Suppose that the conclusion is false. Then there is a rational
number c = α/β, expressed in lowest terms (i.e. α and β have no integer
factors in common) such that c
2
= 2. This translates to
α
2
β
2
=2
or
α
2
=2β
2
.
We conclude that the righthand side is even, hence so is the lefthand
side. Therefore α =2m for some integer m.
But then
(2m)
2

=2β
2
or
2m
2
= β
2
.
So we see that the lefthand side is even, so β is even.
But now both α and β are even—the two numbers have a common
factor of 2. This statement contradicts the hypothesis that α and β have
no common integer factors. Thus it cannot be that c is a rational num-
ber. Instead, c must be irrational.
1.1 Pythagoras 7
For You to Try: Use the argument just presented to show that 7 does
not have a rational square root.
For You to Try: Use the argument just presented to show that if a
positive integer (i.e., a whole number) k has a rational square root then
it has an integer square root.
We stress yet again that the result of the last theorem was a bomb-
shell. It had a profound impact on the thinking of ancient times. For
it established irrefutably that there were new numbers besides the ra-
tionals to which everyone had been wedded. And these numbers were
inescapable: they arose in such simple contexts as the calculation of the
diagonal of a square. Because of this result of Pythagoras, the entire
Greek approach to the number concept had to b e rethought.
1.1.2 Pythagorean Triples
It is natural to ask which triples of integers (a, b, c) satisfy a
2
+ b

2
= c
2
.
Such a trio of numb ers is called a Pythagorean triple.
The most famous and standard Pythagorean triple is (3, 4, 5). But
there are many others, including (5, 12, 13), (7, 24, 25), (20, 21, 29), and
(8, 15, 17). What would be a complete list of all Pythagorean triples?
Are there only finitely many of them, or is there in fact an infinite list?
It has in fact been known since the time of Euclid that there are
infinitely many Pythagorean triples, and there is a formula that generates
all of them.
1
We may derive it as follows. First, we may as well suppose
that a and b are relatively prime—they have no factors in common. We
call this a reduced triple. Therefore a and b are not b oth even, so one of
them is odd. Say that b is odd.
Now certainly (a + b)
2
= a
2
+ b
2
+2ab > a
2
+ b
2
= c
2
. From this we

conclude that c<a+ b. So let us write c =(a + b) −γ for some positive
integer γ. Plugging this expression into the Pythagorean formula ()
1
It may be noted, however, that the ancients did not have adequate notation to write
down formulas as such.
8 Chapter 1: The Ancient Greeks
yields
a
2
+ b
2
=(a + b − γ)
2
or
a
2
+ b
2
= a
2
+ b
2
+ γ
2
+2ab −2aγ − 2bγ .
Cancelling, we find that
γ
2
=2aγ +2bγ − 2ab . (†)
The righthand side is even (every term has a factor of 2), so we conclude

that γ is even. Let us write γ =2m, for m a positive integer.
Substituting this last expression into (†) yields
4m
2
=4am +4bm −2ab
or
ab =2am +2bm −2m
2
.
The righthand side is even, so we conclude that ab is even. Since we have
already noted that b is odd, we can only conclude that a is even. Now
equation ( ) tells us
c
2
= a
2
+ b
2
.
Since the sum of an odd and an even is an odd, we see that c
2
is odd.
Hence c is o dd.
Thus the numbers in a reduced Pythagorean triple are never all even
and never all odd. In fact two of them are odd and one is even. It is
convenient to write b = s −t and c = s + t for some integers s and t (one
of them even and one of them odd). Then () tells us that
a
2
+(s −t)

2
=(s + t)
2
.
Multiplying things out gives
a
2
+(s
2
−2st + t
2
)=(s
2
+2st + t
2
) .
Cancelling like terms and regrouping gives
a
2
=4st .
1.1 Pythagoras 9
We already know that a is even, so this is no great surprise.
Since st must be a perfect square (because 4 is a perfect square and
a
2
is a perfect square), it is now useful to write s = u
2
, t = v
2
. Therefore

a
2
=4u
2
v
2
and hence
a =2uv .
In conclusion, we have learned that a reduced Pythagorean triple
must take the form
(2uv, u
2
− v
2
,u
2
+ v
2
) , (†)
with u, v relatively prime (i.e., having no common factors). Conversely,
any triple of the form (2uv, u
2
−v
2
,u
2
+v
2
) is most certainly a Pythagorean
triple. This may be verified directly:

[2uv]
2
+[u
2
− v
2
]
2
=[4u
2
v
2
]+[u
4
−2u
2
v
2
+ v
4
]
= u
4
+2u
2
v
2
+ v
4
=[u

2
+ v
2
]
2
.
Take a moment to think about what we have discovered. Every
Pythagorean triple must have the form (†). That is to say, a =2uv,
b = u
2
− v
2
, and c = u
2
+ v
2
. Here u and v are any integers of our
choosing.
As examples:
• If we take u = 2 and v = 1 then we obtain a =2·2·1=4,
b =2
2
−1
2
= 3, and c =2
2
+1
2
= 5. Of course (4, 3, 5) is
a familiar Pythagorean triple. We certainly know that

4
2
+3
2
=5
2
.
• If we take u =3andv = 2 then we obtain a =2·3 ·2=
12, b =3
2
− 2
2
= 5, and c =3
2
+2
2
= 13. Indeed
(12, 5, 13) is a Pythagorean triple. We may calculate
that 12
2
+5
2
=13
2
.
• If we take u =5andv = 3 then we obtain a =2·5 ·3=
30, b =5
2
− 3
2

= 16, and c =5
2
+3
2
= 34. You
may check that (30, 16, 34) is a Pythagorean triple, for
30
2
+16
2
=34
2
.
10 Chapter 1: The Ancient Greeks
For You to Try: Find all Pythagorean triples in which one of the
terms is 5.
For You to Try: Find all Pythagorean triples in which all three terms
are less than 30.
1.2 Euclid
1.2.1 Introduction to Euclid
Certainly one of the towering figures in the mathematics of the ancient
world was Euclid of Alexandria (325 B.C.E.–265 B.C.E.). Although Eu-
clid is not known so much (as were Archimedes and Pythagoras) for his
original and profound insights, and although there are not many theo-
rems named after Euclid, he has had an incisive effect on human thought.
After all, Euclid wrote a treatise (consisting of thirteen Books)—now
known as Euclid’s Elements—which has been continuously in print for
over 2000 years and has been through myriads of editions. It is still stud-
ied in detail today, and continues to have a substantial influence over the
way that we think about mathematics.

Not a great deal is known about Euclid’s life, although it is fairly
certain that he had a school in Alexandria. In fact “Euclid” was quite a
common name in his day, and various accounts of Euclid the mathemati-
cian’s life confuse him with other Euclids (one a prominent philosopher).
One appreciation of Euclid comes from Proclus, one of the last of the
ancient Greek philosophers:
Not much younger than these [pupils of Plato] is
Euclid, who put together the Elements, arrang-
ing in order many of Eudoxus’s theorems, per-
fecting many of Theaetus’s, and also bringing to
irrefutable demonstration the things which had
been only loosely proved by his predecessors. This
man lived in the time of the first Ptolemy; for
Archimedes, who followed closely upon the first
1.2 Euclid 11
Ptolemy makes mention of Euclid, and further
they say that Ptolemy once asked him if there
were a shortened way to study geometry than the
Elements, to which he replied that “there is no
royal road to geometry.” He is therefore younger
than Plato’s circle, but older than Eratosthenes
and Archimedes; for these were contemporaries,
as Eratosthenes somewhere says. In his aim he
was a Platonist, being in sympathy with this phi-
losophy, whence he made the end of the whole El-
ements the construction of the so-called Platonic
figures.
As often happens with scientists and artists and scholars of immense
accomplishment, there is disagreement, and some debate, over exactly
who or what Euclid actually was. The three schools of thought are these:

• Euclid was an historical character—a single individual—
who in fact wrote the Elements and the other scholarly
works that are commonly attributed to him.
• Euclid was the leader of a team of mathematicians work-
ing in Alexandria. They all contributed to the creation
of the complete works that we now attribute to Euclid.
They even continued to write and disseminate books
under Euclid’s name after his death.
• Euclid was not an historical character at all. In fact
“Euclid” was a nom de plume—an allonym if you will—
adopted by a group of mathematicians working in Alexan-
dria. They took their inspiration from Euclid of Megara
(who was in fact an historical figure), a prominent philoso-
pher who lived about 100 years before Euclid the math-
ematician is thought to have lived.
Most scholars today subscribe to the first theory—that Euclid was
certainly a unique p erson who created the Elements. But we acknowledge
that there is evidence for the other two scenarios. Certainly Euclid had
12 Chapter 1: The Ancient Greeks
a vigorous school of mathematics in Alexandria, and there is little doubt
that his students participated in his projects.
It is thought that Euclid must have studied in Plato’s Academy in
Athens, for it is unlikely that there would have been another place where
he could have learned the geometry of Eudoxus and Theaetus on which
the Elements are based.
Another famous story and quotation about Euclid is this. A certain
pupil of Euclid, at his school in Alexandria, came to Euclid after learning
just the first proposition in the geometry of the Elements. He wanted
to know what he would gain by putting in all this study, doing all the
necessary work, and learning the theorems of geometry. At this, Euclid

called over his slave and said, “Give him threepence since he must needs
make gain by what he learns.”
What is important about Euclid’s Elements is the paradigm it pro-
vides for the way that mathematics should be studied and recorded. He
begins with several definitions of terminology and ideas for geometry,
and then he records five important postulates (or axioms) of geometry.
A version of these postulates is as follows:
P1 Through any pair of distinct points there passes a line.
P2 For each segment
AB and each segment CD there is a
unique point E (on the line determined by A and B)
such that B is between A and E and the segment CD
is congruent to BE (Figure 1.4(a)).
P3 For each point C and each point A distinct from C there
exists a circle with center C and radius CA (Figure
1.4(b)).
P4 All right angles are congruent.
These are the standard four axioms which give our Eu-
clidean conception of geometry. The fifth axiom, a topic
of intense study for two thousand years, is the so-called
parallel postulate (in Playfair’s formulation):
1.2 Euclid 13
A
B
C
D
E
C
A
l

P
m
(a)
(b)
(c)
Figure 1.4
P5 For each line  and each point P that does not lie on
 there is a unique line m through P such that m is
parallel to  (Figure 1.4(c)).
Of course, prior to this enunciation of his celebrated five axioms,
Euclid had defined point, line, “between”, circle, and the other terms
that he uses. Although Euclid borrowed freely from mathematicians
both earlier and contemporaneous with himself, it is generally believed
that the famous “Parallel Postulate”, that is Postulate P5, is of Euclid’s
own creation.
It should be stressed that the Elements are not simply about geome-
try. In fact Books VII–IX deal with number theory. It is here that Euclid
proves his famous result that there are infinitely many primes (treated
elsewhere in this book) and also his celebrated “Euclidean algorithm” for
long division. Book X deals with irrational numbers, and books XI–XIII
treat three-dimensional geometry. In short, Euclid’s Elements are an
exhaustive treatment of virtually all the mathematics that was known
at the time. And it is presented in a strictly rigorous and axiomatic
manner that has set the tone for the way that mathematics is presented
and studied today. Euclid’s Elements is perhaps most notable for the